Unclear differential equation from a thermodynamics textbook

NODARman
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In the thermodynamics textbook there is written: 𝛿𝐴 = 𝑇𝑑𝑆 βˆ’ π‘‘π‘ˆ = 𝑑(𝑇𝑆) βˆ’ 𝑆𝑑𝑇 βˆ’ π‘‘π‘ˆ = βˆ’π‘‘(π‘ˆ βˆ’ 𝑇𝑆) βˆ’ 𝑆𝑑𝑇 = βˆ’π‘‘πΉ βˆ’ 𝑆𝑑𝑇
How did we get the bolded area from TdS? Is that property of derivative, integral, or something else :/
 
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Product rule
 
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Sorry for being so terse. I was posting from my phone.

Apply the product rule to ##TS##. ##d(TS) = ?##.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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